Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Piotr Sawuk (piotr5_at_unet.univie.ac.at)
Date: 01/18/05 

Date: 18 Jan 2005 04:22:37 GMT

In article <1105674360.668364.215690@f14g2000cwb.googlegroups.com>,
        "Ross A. Finlayson" <raf@tiki-lounge.com> writes:
> In some recent discussion about the well-ordering of sets dense in the
> reals, concepts about the specificities of a "previous" and "next"
> points on the real number line broaden and solidify.
>

> 2. Is it possible to describe and parameterize all possible ways that
> they can be so subdivided?

as you hinted, there are already infinitely many possibilities to divide
rational numbers into sets dense in R: by using only powers of each
prime-number as the denominator. also algorithmically one could divide
real numbers into disjoint sets dense in eachother and in R, and since
uncountabe infinitely many algorithms do exist, so that partition would
contain uncountabe infinitely many such sets of real numbers each dense
within eachother. my guess is that theoretically it would be possible
to create more than alef2 sub-sets of R (such that no bijection does
exist from the set of those sets to the real numbers) together with a
well-ordering such that each such set is dense in all sets with a
higher order of that well-ordering, and each of them is dense within
the set of real numbers. if my guess would reflect the truth (and
thereby prove that indeed all infinities are equal even though no
injective function does exist from alef2 to alef1), how would that
then change your theories?
>

> Under what conditions can it be said the reals are well-orderable? In
> ZFC, any set is well-orderable. If any set of reals dense is
> well-orderable then so is the complete superset and any subset. Is not
> that obvious to you?
>
> If a set is well-orderable, then its elements can be iterated.

of course iterated by ordinal numbers. the whole point of well-ordering
aribatary sets is that even though you could create disjoint subsets
and put some well-ordering on each such subset, you simply can not write
down any algorithm such that found well-ordering does match the order
you already have given on that big set. it's like leaning out of the
window towards infinity: awful things would happen if you fell through,
it's a good thing that you only can see a small part of infinity, it
is a good thing that you can only reach an even smaller part of it, you
really should stop leaning out of that window and rather do something
productive, but finite in nature, instead... 

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